I removed the "propositional calculus" - as you are working with predicate calculus (given your reference in a comment. Unfortunately, there is no tag for predicate-calculus.
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amWhyJan 4 '13 at 16:14

Heavens! Who on earth produced that "model answer"? Complain to your instructor as it really is simply awful -- it really is a straight "fail" answer, I'm afraid.
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Peter SmithJan 5 '13 at 14:27

I'd love to, believe me. But for now this is all I have. However, 1) why is it a "fail" answer? Is it wrong or just badly formed? 2) given that model answer, do you think mine is good?
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4th guyJan 5 '13 at 14:32

The model answer is so far from right that you need to start from scratch. I can only suggest reading a good book on Natural Deduction. The excerpts I can see from Pace's book look OK. But I''d recommend Paul Teller's Modern Formal Logic Primer as reliable, lucid, and now free: you can download chapters from tellerprimer.ucdavis.edu
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Peter SmithJan 5 '13 at 15:12

2 Answers
2

You don't say what/whose system of predicate calculus you are using (textbook reference? the notation is unfamiliar to me), but I know no system which allows
existential quantifier elimination in the form you seem to be using
it.

A normal deduction of this sort of thing, using "introduction" and "elimination" rules, would look more like this:

The notation and the proof-system in that link are quite different from what is displayed in the proofs in the question. The link is to a standard Hilbert system: but it is natural deduction systems that have introduction and elimination rules for quantifiers.
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Peter SmithJan 4 '13 at 16:15

Well, this is the book that we have been referred to: Mathematics of Discrete Structures for Computer Science by Gordon Pace.
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4th guyJan 4 '13 at 17:03

From what's available by way of online excerpts, Pace seems to be using a standard Natural Deduction system, with subproofs (the sort of thing involved in my schematic proof above and conspicuously missing from your suggested proofs).
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Peter SmithJan 4 '13 at 17:35

Your proof looks like a good start, to me, and it is very thorough! Usually, quantifiers are used with predicates, e.g. $$\lnot \forall x.\lnot(P(x)\rightarrow Q(x)),$$ and then, $$\exists x. \lnot\lnot(P(x)\rightarrow Q(x)),$$ etc., so your notation is a bit unconventional.

But I understood well enough that your notation is to be taken as predicates about a quantified $x\in X$, which you (existentially) instantiate some "a" to stand in for "some x", then proceed, until you eliminate the instantiated "a" back to an existentially quantified $x$.

Also, when instantiating an existential quantifier, one usually does so with a "subproof": indented, to help delineate the scope in which it is instantiated.

That said, the logic of your proof follows clearly enough.

Regarding your worries: When you have an equivalence (to prove), the steps for the first implication are sort of (for lack of a better term) "reversed" ("opposite") when proving the second implication. That's partly due to the nature of double implication / equivalence, and the fact that the logical manipulation in your proof (apart from quantifier elimination/introduction) involved only equivalencies (identities).

Nice job of showing your work. And kudos for making the effort to justify all your steps in your proof!

Regarding your Edit: you really ought to include subproofs in your proof: as you are introducing an assumption when you eliminate/instantiate the existentially quantified statement, and need to indicate the scope of that instantiation by indentation, and then end the subproof by reintroducing the existential quantifier (citing the subproof in you justification). Perhaps you can follow the logic here (Note, however, as in Peter Smith's answer, predicates are typically written with an argument (variable or constant):

What I mean is "indented" to show scope of the instantiation: but you can do fine without, it just helps clarify the part of the proof in which you are manipulating an instantiated existential quantifier.
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amWhyJan 5 '13 at 14:43

Unless the solution is wrong, I'd rather not mess with indentation. My head is already hurting by going through all these possible answers.
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4th guyJan 5 '13 at 15:00